Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I

Abstract:

It is natural to extend the Grothendieck Theorem on completeness, valid for locally
convex topological vector spaces, to abelian topological groups. The adequate framework
to do it seems to be the class of locally quasi-convex groups. However, in this paper
we present examples of metrizable locally quasi-convex groups for which the analogue to
Grothendieck Theorem does not hold. By means of the continuous convergence structure
on the dual of a topological group, we also state some weaker forms of Grothendieck
Theorem valid for the class of locally quasi-convex groups. Finally, we prove that for the smaller class of nuclear groups, BB-reflexivity is equivalent to completeness.